曲线乘积的嵌入
Embedding compacta into products of curves
摘要：
我们提出了一些结果，n-维recta躺在n-维产品的recta，特别是，在产品的n 1-维recta。我们的大多数基本结果都是在假设X-球面上有本质映射的前提下得到证明的。本文的结果可以看作是迄今为止由其他作者发展的理论的延伸。首先，证明了如果X是一个具有$H^n(X) \neq 0$的n维紧集，它嵌入n条曲线的乘积或曲线的n阶对称乘积，则存在X到n-环面的代数本质映射.因此，在上积为非零的$H^1(X)$中存在元素$a_1,...,a_n$。因此，$H^1(X)$的秩至少为n，并且X的范畴超过n。其次，我们引入了一些新的n维连续统类，并证明了局部连通拟n维流形嵌入到n条曲线的乘积中也意味着$H^1(X)$的秩至少为n。因此，某些二维可收缩多面体不能嵌入两条曲线的乘积中。另一方面，我们证明了任何可折叠的二维多面体都可以嵌入到两棵树的乘积中。我们回答了一个问题所提出的Cauty证明，闭曲面嵌入产品的两条曲线也可以嵌入产品的两个图。证明了两条曲线的乘积中没有不同于2-环面的闭曲面是该乘积的收缩曲面。
英文摘要：
We present some results on n-dimensional compacta lying in n-dimensional products of compacta, in particular, in products of n 1-dimensional compacta. Most of our basic results are proven under the assumption that the compacta X admit essential maps into the n-sphere. The results of the present paper may be viewed as an extension of the theory developed so far by other authors. First, we prove that if X is an n-dimensional compactum with $H^n(X) \neq 0$ that embeds in either a product of n curves or the nth symmetric product of a curve, then there exists an algebraically essential map of X into the n-torus. Consequently, there exist elements $a_1,...,a_n$ in $H^1(X)$ whose cup product is non-zero. Therefore, the rank of $H^1(X)$ is at least n, and the category of X exceeds n. Next, we introduce some new classes of n-dimensional continua and show that embeddability of locally connected quasi n-manifolds into products of n curves also implies that the rank of $H^1(X)$ is at least n. It follows that some 2-dimensional contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron can be embedded in a product of two trees. We answer a question posed by Cauty proving that closed surfaces embeddable in products of two curves can be also embedded in products of two graphs. We prove that no closed surface, different from the 2-torus, lying in a product of two curves is a retract of that product.
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